[seqfan] Number of unique arrangements of n*n squares

[Joseph Rozhenko]

Hi,

I have an idea for a new sequence.

Given *n* strips, each strip being constituted by *n* squares of unit
length 1 arranged in a single file (strip dimensions are *1*x*n*), what is
number of unique shapes which may be constructed by arranging these strips
side by side while satisfying the condition that each unique shape includes
at least one row of length *n*.

* The term 'unique' refers herein to denote rotational symmetry

For example, for n = 2, there are 3 unique shapes:

*   *        *                 *
*   *   ,    *   *    ,   *   *
                  *        *

For n=3, there are 11 unique shapes (see attached image).

Based on a computer code written for it to check higher numbers, the
results are:

n = 2 --> An = 3
n = 3 --> An = 11
n = 4 --> An = 100
n = 5 --> An = 1,063
n = 6 --> An = 15,686

However, I have no way of verifying the results for n=4, 5 and 6, so I'm
not 100% convinced the code is right.

A more general definition of this problem may be formulated in the
following manner:

Given *n***n* squares of unit length 1, what is the number of unique shapes
which may be formed which satisfies the following two conditions:

1. There is at least one continuous row of squares with a length equal to
*n*; and
2. There are not continuous rows of squares with a length greater than *n*.

I was wondering if anyone might find this interesting enough (or maybe it
has already been looked into by someone).

Thanks,

Yossi Rozhenko